Tedious, but straightforward is just writing out the components.
Try verifying the equation one component a time.
For example, the x-component of A X (B X C) is:

[tex]A_y(B_xC_y-B_yC_x)-A_z(B_zC_x-B_xC_z)[/tex]
, which can be rewritten as:
[tex]B_x(A_yC_y+A_zC_z)-C_x(A_yB_y+A_zB_z)[/tex]

you almost have the x component of B(AC) in the left term and the x-component of C(AB) in the right term. They both miss the same term [itex]B_x(A_xC_x)[/itex] so you can just add it to both terms, since they will cancel. This establishes the relation for the x-component. Do the same for the other two (or try to be smart and use some symmetry argument).